Mathematical Logic – Translation

Translating English sentences into propositional forms

We take an example to explain the procedure:

Example: Convert the following sentence into propositional form:

                “If I earn more than 2Lac Rupees then I must file income tax return”.

1.  Identify the basic simple sentences and denote them with some variables.

       In the above case we can identify two elemental statements:

             I earn more than 2Lac rupees. (Let’s denote it by P)

            I must file income tax return. (Let’s denote it by Q)

 

2.       Look for the connective keywords.

You can identify a number of keywords very easily in any given sentence. These include ‘OR’, ‘AND’, ‘NOT’, ‘IF… THEN’ and ‘Only… IF’. There are, however, two key points to remember and observe:

a.       All keywords except NOT are binary connectives and hence they should be identified only in a manner where they are connecting the two basic statements identified in step 1.

b.      NOT is a unary connective and would be taken as part of a whole statement only, rather than connecting two individual statements.

In the current example, the keywords to note are IF…THEN.

 

3.      Use the identified keywords to represent the given statement in a partially propositional form.

          In the example, such a partial propositional form can be:

                        “If P THEN Q”.

 

4.       Use the proper operator symbols corresponding to keywords now present in the partial form obtained in step 3. For “keyword – connective symbol” relationship you may refer to the previous chapter (INTRODUCTION).

In current example we replace “IF… THEN” with the corresponding arrow symbol ‘→’ to get the final propositional form:

                P → Q.

 

NOTE: While looking for the elemental statements within a given problem statement, identify only affirmative sentences as candidates. Any negation (NOT keyword) should not be taken as part of an elemental statement.

 

More examples:

Translate the following sentences into propositional forms:

(a)     If I earn more than 2Lac Rupees and I am honest, then I will file Income Tax return.

(b)     I do not earn more than 2 Lac rupees and I will not file income tax return.

(c)     I am not honest

(d)     I will not file income tax return only if I am not honest.

 Solution

                Let

                P:   I earn more than 2 Lac Rupees.

                Q:  I am honest.

                R:  I will file income tax return.

Then

(a)     (P Ʌ Q) → R

(b)     ¬P Ʌ ¬R

(c)     ¬Q

(d)     ¬R → ¬P

Rate this post

Leave a Reply